Problem: Simplify the following expression: $ t = \dfrac{5y - 7}{-y - 6} - \dfrac{-2}{3} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{3}{3}$ $ \dfrac{5y - 7}{-y - 6} \times \dfrac{3}{3} = \dfrac{15y - 21}{-3y - 18} $ Multiply the second expression by $\dfrac{-y - 6}{-y - 6}$ $ \dfrac{-2}{3} \times \dfrac{-y - 6}{-y - 6} = \dfrac{2y + 12}{-3y - 18} $ Therefore $ t = \dfrac{15y - 21}{-3y - 18} - \dfrac{2y + 12}{-3y - 18} $ Now the expressions have the same denominator we can simply subtract the numerators: $t = \dfrac{15y - 21 - (2y + 12) }{-3y - 18} $ Distribute the negative sign: $t = \dfrac{15y - 21 - 2y - 12}{-3y - 18}$ $t = \dfrac{13y - 33}{-3y - 18}$ Simplify the expression by dividing the numerator and denominator by -1: $t = \dfrac{-13y + 33}{3y + 18}$